Hello,
This is a general question about the validity of different algorithms for gapless systems and how to get reasonable results from tensor network methods for such systems.
I am trying to find the ground state of a system with a gap that closes at least as fast as \(O(1/N^2)\) for system size N. I am trying two-site DMRG at the moment, with a decaying subspace expansion mixer strength. I find that I receive the "FInal state not in canonical form warning" during many of my runs, which I am taking as a hint that DMRG is not converging for the given bond dimension. For a system size of L=8 with subspace expansion turned on, I find that a bond dimension of 256 is enough for convergence. Clearly, larger systems will need strong truncation of some kind. Do you have general advice on getting reasonable ground state approximations for such systems with fast vanishing gaps using Matrix product states? I considered using VUMPS, however, the Hamiltonian for the model I am considering has boundary projectors that enforce boundary conditions on edge spins - I am not entirely sure how this would translate to a uniform MPS Ansatz or if this even makes sense to do. Apologies if my question is too vague - in short, I am wondering if there are known strategies for dealing with gapless / vanishing gap systems with tensor networks for medium (~L=32/64) finite size systems.
Thank you!
Finding ground states of finite systems with fast vanishing gap
Re: Finding ground states of finite systems with fast vanishing gap
When you need chi=256 for L=8, you apparently have local hilbert space dimension of 4 - is it a spin-full fermion site? Do you have any charge conservation/symmetries in general?
DMRG should usually be able to converge L=32/64 pretty easily.
One issue you can have though are close to exact degeneracies, where DMRG might swap between different ground states during the sweep - given that you have a quickly vanishing gap, maybe this is the issue?
In the simplest example, the TFI model in the symmetry broken phase, the (dressed) all-up and all-down state are nearly degenerate - for finite system size, the exact ground states are actually (anti-)symmetric superpositions of those, but as you increase system size, the degeneracy falls below machine precision/error tolerance/truncation error, and you can no longer resolve it. Then DMRG typically "selects" a superposition of the possible solution with little entanglement, say the all-up state, because it needs less truncation for that (and thous can improve the energy a little bit further).
In your case, is it only a single state that comes down closing the gap, or is it a whole continuum?
Can you distinguish the states by bulk expectation values, or are it only edge modes? If it's only edge modes, the DMRG in the bulk could randomly jump between solutions, explaining the poor convergence.
One thing that you could try then is to lift the edge mode degeneracy, either with explicit terms in you Hamiltonian (that you might try to switch off after an initial convergence?), or more extremley by finding one solution with DMRG (calling DMRG.canonical_form() in the end) and the using the `orthogonal_to` DMRG parameter to get the other, nearly degenerate solution.
DMRG should usually be able to converge L=32/64 pretty easily.
One issue you can have though are close to exact degeneracies, where DMRG might swap between different ground states during the sweep - given that you have a quickly vanishing gap, maybe this is the issue?
In the simplest example, the TFI model in the symmetry broken phase, the (dressed) all-up and all-down state are nearly degenerate - for finite system size, the exact ground states are actually (anti-)symmetric superpositions of those, but as you increase system size, the degeneracy falls below machine precision/error tolerance/truncation error, and you can no longer resolve it. Then DMRG typically "selects" a superposition of the possible solution with little entanglement, say the all-up state, because it needs less truncation for that (and thous can improve the energy a little bit further).
In your case, is it only a single state that comes down closing the gap, or is it a whole continuum?
Can you distinguish the states by bulk expectation values, or are it only edge modes? If it's only edge modes, the DMRG in the bulk could randomly jump between solutions, explaining the poor convergence.
One thing that you could try then is to lift the edge mode degeneracy, either with explicit terms in you Hamiltonian (that you might try to switch off after an initial convergence?), or more extremley by finding one solution with DMRG (calling DMRG.canonical_form() in the end) and the using the `orthogonal_to` DMRG parameter to get the other, nearly degenerate solution.